Unimodal sequences and “strange” functions: a family of quantum modular forms

Bringmann, Kathrin; Folsom, Amanda; Rhoades, Robert C.

doi:10.2140/pjm.2015.274.1

Cited by 21 publications

(41 citation statements)

References 25 publications

(21 reference statements)

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“…5 for more details.) The theory of quantum modular forms is in its beginning stages; constructing explicit examples of these functions remains of interest, as does answering the question of how quantum modular forms may arise from mock modular forms (see the recent articles [3,6,13], for example).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Quantum mock modular forms arising from eta–theta functions

et al. 2016

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In 2013, Lemke Oliver classified all eta-quotients which are theta functions. In this paper, we unify the eta-theta functions by constructing mock modular forms from the eta-theta functions with even characters, such that the shadows of these mock modular forms are given by the eta-theta functions with odd characters. In addition, we prove that our mock modular forms are quantum modular forms. As corollaries, we establish simple finite hypergeometric expressions which may be used to evaluate Eichler integrals of the odd eta-theta functions, as well as some curious algebraic identities.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…3 We first establish (8) and (10). To do so, we begin with parts (i) and (ii) of Theorem 1.2 in the case n = 1.…”

Section: Corollariesmentioning

confidence: 99%

Quantum mock modular forms arising from eta–theta functions

et al. 2016

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“…Remark 1.3. Considering the catalog of functions V mn from Folsom et al [9], we see that by choosing (A, C, a) from the set {(1, 4, 1), (1,4,0), (1,3,1), (1,12,0), (5,12,0), (1,6,1), (1, 3, 0)}, V α yields the functions V 11 , V 21 , V 31 , V 4 ′ 1 , V 4 ′′ 1 , V 51 , V 61 . In particular, we note that we can generate the first row of each table in [9] using the V α function.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Extending a catalog of mock and quantum modular forms to an infinite class

Arnold-Roksandich

Diaz²,

Ellefsen

et al. 2020

Res. number theory

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Utilizing a classification due to Lemke Oliver of eta-quotients which are also theta functions (here called eta-theta functions), Folsom, Garthwaite, Kang, Treneer, and the fourth author constructed a catalog of mock modular forms Vmn having weight 3/2 eta-theta function shadows and showed that these mock modular forms when viewed on certain sets of rationals, transform as quantum modular forms under the action of explicit subgroups. In this paper, we introduce an infinite class of functions that generalizes one row of the catalog, namely the V m1 , and show that the functions in this infinite class are both mock modular and quantum modular forms.where τ is in the upper half-plane H = {τ ∈ C | Im(τ )}, and q = e 2πiτ is one of the most well-known half-integral weight modular forms. Quotients of the form c j=1 η(a j τ ) bj , for b j ∈ Z are called eta-quotients, and have been of interest in many areas of mathematics, including combinatorics, number theory, and representation theory (see [16], [1], [7], for example). Classifications of eta-quotients have also been of interest. Dummit, Kisilevsky, and McKay [8] classified all multiplicative eta-products (restricting to positive b j ), and later Martin [14] classified multiplicative integer weight eta-quotients. Furthering these classifications, Lemke Oliver [15] recently classified all eta-quotients which are also theta functions, which are of the form θ χ (τ ) := n χ(n)n ν q n 2 ,where χ is an even (resp. odd) Dirichlet character, ν is 0 (resp. 1), and the sum above is over n ∈ Z when χ is trivial, or over n ∈ N when χ is not trivial. These functions are known to be ordinary modular forms of weight 1/2 + ν.We refer to a function which is both an eta-quotient and a theta function as an eta-theta function. In Lemke Oliver's classification there are six odd (character) eta-theta functions E m , and eighteen even (character) eta-theta functions e n (some of which are twists by certain principal characters).Modular theta functions also arise in the theory of harmonic Maass forms and mock modular forms via the shadow operator. A harmonic Maass form f , as originally defined by Bruinier and Funke [5], naturally decompose into two parts as f = f + + f − , where f + is called the holomorphic part of f , and f − is called the non-holomorphic part of f . All of Ramanujan's original mock theta functions turn out to be examples of holomorphic parts of harmonic Maass forms [20], and we now define after Zagier [18] a mock modular form to be any such holomorphic part of a harmonic Maass form (for which the non-holomorphic part is not

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“…For example, the connection between them and mock modular forms (surveyed in e.g. [19]) is investigated in papers such as [4,9,10], among others. Furthermore, interesting examples of quantum modular forms exist in the interface of physics and knot theory, see e.g.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

A family of vector-valued quantum modular forms of depth two

Males

2019

Int. J. Number Theory

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We introduce and investigate an infinite family of functions which are shown to have generalised quantum modular properties. We realise their "companions" in the lower half plane both as double Eichler integrals and as non-holomorphic theta functions with coefficients given by double error functions. Further, we view these Eichler integrals in a modular setting as parts of certain weight two indefinite theta series.

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Unimodal sequences and “strange” functions: a family of quantum modular forms

Cited by 21 publications

References 25 publications

Quantum mock modular forms arising from eta–theta functions

Quantum mock modular forms arising from eta–theta functions

Extending a catalog of mock and quantum modular forms to an infinite class

A family of vector-valued quantum modular forms of depth two

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